[Paper Review] On total dominating sets in graphs

Maryam Atapour, Nasrin Soltankhah|ArXiv.org|Oct 26, 2008

Advanced Graph Theory Research2 references3 citations

TL;DR

This paper characterizes bipartite graphs and trees that achieve the upper bound $\gamma_t(G) \leq n - \Delta(G) + 1$ for the total domination number, proving that such graphs are disjoint unions of a star $K_{1,t}$ and $r$ copies of $K_2$. It also establishes new upper and lower bounds for $\gamma_t(G)$, and determines the exact total domination number for circular complete graphs $K_{n,d}$, showing $\gamma_t(K_{n,d}) = 2$ when $n \geq 4d-2$ and $\gamma_t(K_{n,d}) = 3$ when $3d \leq n \leq 4d-3$. The results refine known bounds and provide exact values for a class of dense graphs.

ABSTRACT

A set $S$ of vertices in a graph $G(V,E)$ is called a dominating set if every vertex $v\in V$ is either an element of $S$ or is adjacent to an element of $S$. A set $S$ of vertices in a graph $G(V,E)$ is called a total dominating set if every vertex $v\in V$ is adjacent to an element of $S$. The domination number of a graph $G$ denoted by $γ(G)$ is the minimum cardinality of a dominating set in $G$. Respectively the total domination number of a graph $G$ denoted by $γ_t(G)$ is the minimum cardinality of a total dominating set in $G$. An upper bound for $γ_t(G)$ which has been achieved by Cockayne and et al. in $\cite{coc}$ is: for any graph $G$ with no isolated vertex and maximum degree $Δ(G)$ and $n$ vertices, $γ_t(G)\leq n-Δ(G)+1$. Here we characterize bipartite graphs and trees which achieve this upper bound. Further we present some another upper and lower bounds for $γ_t(G)$. Also, for circular complete graphs, we determine the value of $γ_t(G)$.

Motivation & Objective

• To characterize all bipartite graphs and trees that achieve the upper bound $\gamma_t(G) \leq n - \Delta(G) + 1$ for the total domination number.
• To establish new upper and lower bounds for $\gamma_t(G)$ in general graphs.
• To determine the exact value of the total domination number for circular complete graphs $K_{n,d}$.

Proposed method

• Proved that $\gamma_t(G) \geq \lceil n / \Delta(G) \rceil$ using degree constraints on total dominating sets.
• Used diameter and girth conditions to derive bounds: $\gamma_t(G) \leq \delta(G) + 1$ for diameter 2 graphs, and $\gamma_t(G) \leq n - \lceil g(G)/2 \rceil + 1$ for graphs with girth $\geq 5$.
• Employed structural analysis on bipartite graphs with partitions $A$ and $B$, focusing on vertex degrees and neighborhood relations to derive necessary conditions for achieving the upper bound.
• Used contradiction and construction techniques to show that only graphs of the form $K_{1,t} \cup rK_2$ achieve the bound in bipartite graphs.
• Constructed explicit total dominating sets for circular complete graphs $K_{n,d}$, proving $\gamma_t = 2$ for $n \geq 4d-2$ and $\gamma_t = 3$ for $3d \leq n \leq 4d-3$.
• Verified sharpness of bounds using extremal examples such as $K_n$, $C_{4n}$, $P_{4n}$, and $C_5$.

Experimental results

Research questions

• RQ1 Which bipartite graphs satisfy $\gamma_t(G) = n - \Delta(G) + 1$?
• RQ2 Which trees satisfy $\gamma_t(T) = n - \Delta(T) + 1$?
• RQ3 What are tight upper and lower bounds for $\gamma_t(G)$ in terms of $n$, $\Delta(G)$, $\delta(G)$, and girth?
• RQ4 What is the exact value of $\gamma_t(K_{n,d})$ for circular complete graphs?
• RQ5 For which $n$ and $d$ does $\gamma_t(K_{n,d}) = 2$ or $3$?

Key findings

• The only bipartite graphs achieving $\gamma_t(G) = n - \Delta(G) + 1$ are those of the form $K_{1,t} \cup rK_2$ for $r \geq 0$, where $t = \Delta(G)$.
• The only trees achieving $\gamma_t(T) = n - \Delta(T) + 1$ are stars, i.e., $T = K_{1,n-1}$.
• For any connected graph $G$, $\gamma_t(G) \geq \lceil n / \Delta(G) \rceil$, and this bound is sharp for $K_n$, $C_{4n}$, and $P_{4n}$.
• If $\text{diam}(G) = 2$, then $\gamma_t(G) \leq \delta(G) + 1$, and this bound is tight for $C_5$, where $\gamma_t(C_5) = 3 = \delta(C_5) + 1$.
• For connected graphs with girth $g(G) \geq 5$ and $\delta(G) \geq 2$, $\gamma_t(G) \leq n - \lceil g(G)/2 \rceil + 1$, with equality possible under specific structural conditions.
• For circular complete graphs $K_{n,d}$ with $d \geq 3$, $\gamma_t(K_{n,d}) = 2$ when $n \geq 4d - 2$, and $\gamma_t(K_{n,d}) = 3$ when $3d \leq n \leq 4d - 3$.